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Let be an open set of a Stein manifold of dimension such that for . We prove that is Stein if and only if every topologically trivial holomorphic line bundle on is associated to some Cartier divisor on .
Let D be an open subset of a two-dimensional Stein manifold S. Then D is Stein if and only if every holomorphic line bundle L on D is the line bundle associated to some (not necessarily effective) Cartier divisor 𝔡 on D.
We establish the homotopy classification of holomorphic submersions from Stein manifolds
to Complex manifolds satisfying an analytic property introduced in the paper. The result
is a holomorphic analogue of the Gromov--Phillips theorem on smooth submersions.
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